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‘What a difference a day makes…’: Concern about a new approach to valuing a life
JOURNALOF ELSEV1EK
PUBLIC ECONOMICS
Journal of Public Economics 61 (1996) 455-457
' W h a t a difference a day m a k e s . . . ' : C o n c e r n a b o u t a new a p p r o a c h to valuing a life John G. Cullis, Philip R. Jones* School 0I" Social Sciences. University of Bath. Claverton Down. Bath BA2 7.4 Y. UK
Received September 1993; final version received June 1995
Abstract If individuals are interested in life per se (or perceive longevity as a goal worth attaining), there is reason to doubt that Brent (Journal o f Public Economica, 1991, 22, 165-171) has provided a robust solution to the valuation of life in cost-benefit analysis. Keywords: Value of life; Safety; Cost-benefit analysis J E L classification: J17; J28
Brent (1991) attempts to value life by using time as a n u m e r a i r e ) In period 1 an individual has 'time available" which can be used to produce income and consumption directly or used for life safety (S). Life safety is not required in and of itself but only because, by devoting time to S, " . , . i t is expected that there is more time available in period 2, and thereby more f u t u r e earnings a n d c o n s u m p t i o n . . . " (p. 167, our emphasis). The individual must allocate time in period 1 (Tt), i.e. the individual is granted an e n d o w m e n t of certain time in period 1. Individuals who maximise the present value of lifetime income (consumption) apply a neat 'one for one' rule:
(1)
dP(S)/dS. T,/1 + r = 1,
* Corresponding author. Tel.: + 01225 826826; fax: p.r.jones(dbath.ac.uk. ' In doing so. he follows Dowie (1970).
+01225 826381; e-mail:
0047-2727/9(~/gi5.00 ~) 1996 Elsevier Science S.A. All rights reserved S S D ! 0047-2727(95)01550-7
456
J.G. Cullis. P.R. Jones / Journal of Public Economics 61 (1996) 455-457
where P is the probability of surviving; T_, is the future time: r is "the relevant" discount rate: and S is the units of time in period 1 devoted to safety. The advantage claimed for using time as numeriare is that it avoids a m o n e t a r y valuation of human life. Brent illustrates its potential by reference to a U.S. speed limit decision. Aggregate years of life saved by a reduction iv_ accidents are compared with years lost (in aggregate) by driving more slowly. Since "(t)he undiscounted net benefit in terms of years of life w a s - 139,721...(t)he m a x i m u m benefit-cost ratio in terms of time would have been 0.69"" (p. 169, author's emphasis) a decision against the speed limit can be made without evaluating life in monetary terms. Brent's rule is derived from "' ..the individual's lifetime budget const.raint...'" (p. 168),-" where the benefit of safety time, B(S), depends upon consumption possibilities in the first period, i.e. w ( T 1 - S ) . and discounted cxpected consumption in the second period, i.e. wElT2]~(1 + r), where w is the constant wage rate. The individual is assumed to maximise the present value of income: w ( T l - S) + w P ( S ) T z / 1 + r .
(2)
Brent's rule is invalid if an individual is interested in longevity and is prepared to sacrifice a day now for, say, eight hours (discounted) in some future time period. If life is valued for itself, the probability of 'being around" in the second period matters and the maximand becomes U-~U{C'P(S)}=U{[
w'(T'-S)+
w P ( S ) . T 2]; P ( S ) } , 2 l+r J
(3)
where C refers to the two-nerlc~ci, . . . . . consumption possibilities, wl and w, are the first- and second-period wage rates, respectively, and other terms are as defined previously. Maximising (3) with respect to S and rearranging yields: OP(S) T: OS 1 + r From (a) (b) when
w~ w,
OU OC OP(S) OP(S) OU OS
1 w,_
(4)
Eq. (4), two of a n u m b e r of possible cases are: Brent's rule (Eq. (1)) arises when w~ = w, and OU/OP(S) = 0. The more typical life-valued case (as represented in Eq. (3)) arises w~ = w, and ,gU/OP(S)~O in Eq. (4). so that OP(S)/OS. T , / 1 + r < 1. _
(5)
-"Brent refers to the equation as a "budget constraint', but his derived rule would represent a very special case if it were interpreted as such. He also uses the equation as a maximand and accepts that this depends on a constraining assumption, i.e. of the consumer "'...maximizingthe present value of lifetime wage income" (p. 168).
J.G. Cullis. PR. Jones / Journal of Public Economics 61 (1996) 455-457
457
In short, even with a constant wage rate only in the limiting special case w h e n O U / ~ P ( S ) = 0 does Brent's rule e m e r g e . T o use this shadow price in c o s t - b e n e f i t analysis is at odds with the observation that, for many people, t h e r e is no current period consumption level that will c o m p e n s a t e for the certainty of being d e a d in the next period. 4 Life itself is a prize worth having. Sen (1973) offers an insight into issues o f equity by posing the question: Would you willingly trade places with a n o t h e r specified individual? T h e relevant g e d a n k e n question here is: Would y o u want a life (saving) policy decision affecting you to be evaluated by r e f e r e n c e to ' B r e n t ' s rule'? O u r response is 'no'.
Acknowledgements T h e a u t h o r s wish to thank J o h n B r o o m e , David Collard, the editors o f this journal, and two a n o n y m o u s referees for helpful c o m m e n t s and encouragem e n t with this contribution. T h e authors alone remain responsible for any errors.
References Brent, R.J.. 1991. A new approach to valuing a life, Journal of Public Economics 22. 165-171. Dowie. J.A., 1970,Valuing the benefits of health improvement. Australian Economic Papers 9. 12-41. Sen. A., 1973. On economic inequality (Clarendon Press. Oxford). The analysis can be extended by considering the situation in which wage rates change as between periods 1 and 2. Although. in this analysis such an observation is not at odds with a willingness to accept a reduction in the probability of survival for some additional present consumption.
PUBLIC ECONOMICS
Journal of Public Economics 61 (1996) 455-457
' W h a t a difference a day m a k e s . . . ' : C o n c e r n a b o u t a new a p p r o a c h to valuing a life John G. Cullis, Philip R. Jones* School 0I" Social Sciences. University of Bath. Claverton Down. Bath BA2 7.4 Y. UK
Received September 1993; final version received June 1995
Abstract If individuals are interested in life per se (or perceive longevity as a goal worth attaining), there is reason to doubt that Brent (Journal o f Public Economica, 1991, 22, 165-171) has provided a robust solution to the valuation of life in cost-benefit analysis. Keywords: Value of life; Safety; Cost-benefit analysis J E L classification: J17; J28
Brent (1991) attempts to value life by using time as a n u m e r a i r e ) In period 1 an individual has 'time available" which can be used to produce income and consumption directly or used for life safety (S). Life safety is not required in and of itself but only because, by devoting time to S, " . , . i t is expected that there is more time available in period 2, and thereby more f u t u r e earnings a n d c o n s u m p t i o n . . . " (p. 167, our emphasis). The individual must allocate time in period 1 (Tt), i.e. the individual is granted an e n d o w m e n t of certain time in period 1. Individuals who maximise the present value of lifetime income (consumption) apply a neat 'one for one' rule:
(1)
dP(S)/dS. T,/1 + r = 1,
* Corresponding author. Tel.: + 01225 826826; fax: p.r.jones(dbath.ac.uk. ' In doing so. he follows Dowie (1970).
+01225 826381; e-mail:
0047-2727/9(~/gi5.00 ~) 1996 Elsevier Science S.A. All rights reserved S S D ! 0047-2727(95)01550-7
456
J.G. Cullis. P.R. Jones / Journal of Public Economics 61 (1996) 455-457
where P is the probability of surviving; T_, is the future time: r is "the relevant" discount rate: and S is the units of time in period 1 devoted to safety. The advantage claimed for using time as numeriare is that it avoids a m o n e t a r y valuation of human life. Brent illustrates its potential by reference to a U.S. speed limit decision. Aggregate years of life saved by a reduction iv_ accidents are compared with years lost (in aggregate) by driving more slowly. Since "(t)he undiscounted net benefit in terms of years of life w a s - 139,721...(t)he m a x i m u m benefit-cost ratio in terms of time would have been 0.69"" (p. 169, author's emphasis) a decision against the speed limit can be made without evaluating life in monetary terms. Brent's rule is derived from "' ..the individual's lifetime budget const.raint...'" (p. 168),-" where the benefit of safety time, B(S), depends upon consumption possibilities in the first period, i.e. w ( T 1 - S ) . and discounted cxpected consumption in the second period, i.e. wElT2]~(1 + r), where w is the constant wage rate. The individual is assumed to maximise the present value of income: w ( T l - S) + w P ( S ) T z / 1 + r .
(2)
Brent's rule is invalid if an individual is interested in longevity and is prepared to sacrifice a day now for, say, eight hours (discounted) in some future time period. If life is valued for itself, the probability of 'being around" in the second period matters and the maximand becomes U-~U{C'P(S)}=U{[
w'(T'-S)+
w P ( S ) . T 2]; P ( S ) } , 2 l+r J
(3)
where C refers to the two-nerlc~ci, . . . . . consumption possibilities, wl and w, are the first- and second-period wage rates, respectively, and other terms are as defined previously. Maximising (3) with respect to S and rearranging yields: OP(S) T: OS 1 + r From (a) (b) when
w~ w,
OU OC OP(S) OP(S) OU OS
1 w,_
(4)
Eq. (4), two of a n u m b e r of possible cases are: Brent's rule (Eq. (1)) arises when w~ = w, and OU/OP(S) = 0. The more typical life-valued case (as represented in Eq. (3)) arises w~ = w, and ,gU/OP(S)~O in Eq. (4). so that OP(S)/OS. T , / 1 + r < 1. _
(5)
-"Brent refers to the equation as a "budget constraint', but his derived rule would represent a very special case if it were interpreted as such. He also uses the equation as a maximand and accepts that this depends on a constraining assumption, i.e. of the consumer "'...maximizingthe present value of lifetime wage income" (p. 168).
J.G. Cullis. PR. Jones / Journal of Public Economics 61 (1996) 455-457
457
In short, even with a constant wage rate only in the limiting special case w h e n O U / ~ P ( S ) = 0 does Brent's rule e m e r g e . T o use this shadow price in c o s t - b e n e f i t analysis is at odds with the observation that, for many people, t h e r e is no current period consumption level that will c o m p e n s a t e for the certainty of being d e a d in the next period. 4 Life itself is a prize worth having. Sen (1973) offers an insight into issues o f equity by posing the question: Would you willingly trade places with a n o t h e r specified individual? T h e relevant g e d a n k e n question here is: Would y o u want a life (saving) policy decision affecting you to be evaluated by r e f e r e n c e to ' B r e n t ' s rule'? O u r response is 'no'.
Acknowledgements T h e a u t h o r s wish to thank J o h n B r o o m e , David Collard, the editors o f this journal, and two a n o n y m o u s referees for helpful c o m m e n t s and encouragem e n t with this contribution. T h e authors alone remain responsible for any errors.
References Brent, R.J.. 1991. A new approach to valuing a life, Journal of Public Economics 22. 165-171. Dowie. J.A., 1970,Valuing the benefits of health improvement. Australian Economic Papers 9. 12-41. Sen. A., 1973. On economic inequality (Clarendon Press. Oxford). The analysis can be extended by considering the situation in which wage rates change as between periods 1 and 2. Although. in this analysis such an observation is not at odds with a willingness to accept a reduction in the probability of survival for some additional present consumption.